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Abstract

A convolution-backprojection formula is deduced for direct reconstruction of a three-dimensional density function from a set of two-dimensional projections. The formula is approximate but has useful properties, including errors that are relatively small in many practical instances and a form that leads to convenient computation. It reduces to the standard fan-beam formula in the plane that is perpendicular to the axis of rotation and contains the point source. The algorithm is applied to a mathematical phantom as an example of its performance.

R. M. Lewitt and M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

R. M. Lewitt and M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

McKay, M. R.

R. M. Lewitt and M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

R. M. Lewitt and M. R. McKay, “Description of a software package for computing cone-beam x-ray projections of time-varying structures, and for dynamic three-dimensional image reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., May, 1980).

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” (State University of New York at Buffalo, Buffalo, N.Y., June, 1981).

Fig. 2 Geometry in the midplane for derivation of the fan-beam formula. The detector system is here represented by its projection on a line (a–a′) through the origin and parallel to the actual detector line.

Fig. 4 Coordinate system for projections above the midplane. The axis of rotation is along z. The vector
n^ is parallel to the midplane. The vector
k^ is inclined with respect to z and is given by
k^=m^×n^. ρ′ lies in the shaded plane.

Fig. 5 Comparison of representative slices of a phantom with its reconstruction. The phantom and the detector array used are defined in Table 1. The lower row of slices is an exact digitized representation of the phantom, with the corresponding reconstruction just above. The horizontal line defines the position corresponding to the line drawing, in which the density of the phantom (solid line) is compared with that of the reconstruction (points). The scale is linear with a range of 0.0 to 1.0.

Fig. 7 Comparison of vertical (x = constant) slices of the exact phantom (middle row) with corresponding slices from the present method (row below middle) and the unmodified fan-beam method (row above middle). The display scale has been concentrated in the density range 0.64 to 1.44 for clarity. Absolute differences from exact are shown in the bottom row (present method) and the top row (unmodified fan-beam method). In order to highlight small differences, the scale is linear from 0.0 to 0.2. The phantom contains the two additional ellipsoids noted in Table 1.

a (After Ref. 7.) The phantom is constructed by superposing the density contributions of six ellipsoids. The first two have been elongated to generate an inner cylinder and an outer cylindrical shell. Ellipsoids 7 and 8 apply to Figs. 7 and 8 only.